Minimizing nets and generalized best approximation elements

A concept of a generalized best approximation element (GBA) with respect to arbitrary sets is introduced. Efimov–Stechkin spaces are those in which every GBA is in fact some usual best approximation element. Some properties of minimizing nets are studied. The separation of sets from balls by finitely many hyperplanes is also considered. An example of a smooth Banach space is given, in which there exists a nonconvex quasi-Chebyshev set (i.e. a set with respect to which for every $x\in X$ there exists a unique GBA).